Efficient Ray Tracing Through Aspheric Lenses and Imperfect

Home , Lens flare, Simplicity (photography)

Eurographics Symposium on Rendering 2016 Volume 35 (2016), Number 4 E. Eisemann and E. Fiume (Guest Editors)

Efﬁcient Ray Tracing Through Aspheric Lenses and Imperfect Bokeh Synthesis

Hyuntae Joo1, Soonhyeon Kwon1, Sangmin Lee1, Elmar Eisemann2 and Sungkil Lee†1

1Sungkyunkwan University, South Korea 2Delft University of Technology, Netherlands

Figure 1: Examples of bokeh textures generated with our ray-tracing solution.

Abstract We present an efficient ray-tracing technique to render bokeh effects produced by parametric aspheric lenses. Contrary to conventional spherical lenses, aspheric lenses do generally not permit a simple closed-form solution of ray-surface intersections. We propose a numerical root-finding approach, which uses tight proxy surfaces to ensure a good initialization and convergence behavior. Additionally, we simulate mechanical imperfections resulting from the lens fabrication via a texture-based approach. Fractional Fourier transform and spectral dispersion add additional realism to the synthesized bokeh effect. Our approach is well-suited for execution on graphics processing units (GPUs) and we demonstrate complex defocus-blur and lens-flare effects.

Categories and Subject Descriptors (according to ACM CCS): I.3.3 [Computer Graphics]: Picture/Image Generation—Display Algorithms

1. Introduction no general closed-form solution for calculating ray-surface intersections, as exists for spherical lenses [KMH95, SDHL11, HESL11]. Optical systems are typically imperfect and not all rays will follow Instead, we present an efficient numerical scheme to trace rays the expected optical path toward the sensor, which leads to optical through these lens elements. Additionally, aspheric lenses often con- aberrations [Smi00]. Classical systems make use of many spherical tain small imperfections due to the fabrication process (e.g., mechan- lens elements to reduce this effect. A recent alternative is an aspheric ical grinding [LB07]), influencing out-of-focus blur. In particular, it lens, whose profile does not correspond to a part of a sphere [KT80]. can impact the intensity profile of bokeh, the aesthetic appearance of Their use typically implies fewer optical elements for a similar out-of-focus highlights, resulting in “onion-ring bokeh” [ST14]. We reduction of aberrations, which, consequently, results in less weight. simulate this characteristic aspect by using a normal map derived This property made them a popular choice (e.g., smartphones lenses, from a virtual grinding process. pancake lenses, and eyeglasses). Despite their wide-spread use in practice, aspheric lenses have hardly been investigated in computer The contributions of this paper can be summarized as: graphics. • an aspheric lens model, including imperfections; • In this paper, we model and simulate aspheric lenses. Because an efficient numerical aspheric-lens intersection method; their surface profile is usually nonlinear and non-spherical, there is • a rendering solution for optical systems. Our paper is organized as follows. We discuss previous work (Sec.2) and introduce our aspheric lens model (Sec.3), including † [email protected] (corresponding author) imperfections (Sec.4). We then cover our rendering algorithm and

© 2016 The Author(s) Computer Graphics Forum © 2016 The Eurographics Association and John Wiley & Sons Ltd. Published by John Wiley & Sons Ltd. H. Joo & S. Kwon & S. Lee & E. Eisemann & S. Lee / Efﬁcient Ray Tracing Through Aspheric Lenses and Imperfect Bokeh Synthesis intersection test for aspheric lenses (Sec.5) before presenting results sa sa (Sec.6) and concluding (Sec.7).

(a) spherical lens (b) aspheric lens 2. Related Work This section reviews previous work on simulating optical systems Figure 2: A spherical lens (a) can cause spherical aberration, while and ray-tracing techniques for parametric surfaces. even a single aspheric lens (b) may focus well.

2.1. Optical Simulation but unreliable for complex surfaces. These problems manifest them- Early approaches for optical simulations have relied on simple op- selves even more clearly when using limited precision floating-point tical models [Coo86, Che87], but simulating a full optical system numbers, as is often the case for GPU approaches. In contrast, our has been of growing interest. Many approaches opted for additional solution relies on a bracketing method supported by tight proxy realism in terms of geometric distortion, defocus blur, and spectral geometry to perform the intersection test, which even works for dispersion. They usually build upon formal data sheets specifying lower-precision floating point values. the optical system in terms of its optical elements. The seminal work of Kolb et al. obtained geometric accuracy for distortion and field of view [KMH95]. Steinert et al. included additional aberrations and 3. Aspheric-Lens Model spectral diversities using Monte-Carlo rendering [SDHL11]. Hullin We here introduce our parametric aspheric-lens model. An aspheric et al. added reflections and anti-reflective coatings, which play a role lens adds nonlinear deviations to its base curvature c of a spherical for lens flares [HESL11]. They attained physically faithful image surface (Figure2), leading to a better convergence of peripheral rays quality but were restricted to spherical lenses in order to simplify on the focus plane. The common parametric form of an aspheric the ray-surface intersection tests. In this paper, we add support for surface with its sagittal profile z on the optical axis (along the z-axis), aspheric lenses. is expressed in terms of the radial coordinate r from the optical axis Optical imperfections can be classified into intrinsic and extrin- as: 2 sic ones. The former ones include aberrations and flares [HESL11, cr 2i z(r) := p + ∑A2ir , (1) LE13], the latter ones include issues due to mechanical manufactur- 1 + 1 − (1 + κ)c2r2 ing (e.g., grinding and polishing) and real-world artifacts (e.g., dirt or axis misalignment). where κ is a conic constant [FTG00] and A2i are coefficients for even-polynomial sections; the coefficients of particular systems can One of the pronounced intrinsic imperfections are (spherical) be found in patents or lens design books [Smi05]. aberrations manifesting in bokeh and can be simulated via ray tracing [LES10, WZH∗10, WZHX13]. For higher performance, early The design of an aspheric lens goes through an iterative opti- models captured the appearance phenomenologically [BW02]. Later mization procedure to minimize aberrations. The conic constant approaches relied on efficient filtering; polygonal filtering [MRD12], κ is often a consequence of the lens’ base shape from which the separable filtering [MH14], low-rank linear filters [McG15], or look- aspheric lens is derived, hence, the focus lies on optimizing A2i. up table [GKK15]. The design requires high expertise and care in optics. We refer the reader to the textbooks for mathematical properties and practical Extrinsic imperfections are not easily incorporated into ideal para- usages [FTG00, Smi05]. metric models [KMH95,WZH ∗10,SDHL11,HESL11,WZHX13]. One attempt was to use artificial noise for diffraction at the aperture [RIF∗09,HESL11], whereas our method is driven by simulating 4. Imperfections the lens-fabrication process. In the real world, the fabrication of large spherical/aspheric lenses involves mechanical grinding and polishing [LB07] (Figure3). The process has usually several phases; for instance, spherical grinding, 2.2. Ray Tracing of Parametric Surfaces precision grinding, and polishing are applied in sequence [FTG00]. While many modern ray tracers focus on polygonal mod- Similarly, in precision glass molding, a mold is carved, which will els [PBMH02,FS05], early approaches often included several para- then exhibit the imperfections that are passed on to the lens, when it metric surface types, which can be classified in four categories: sub- is produced with the mold. The processes are not entirely standard- division [Whi80], algebraic methods [Kaj82,Bli82, Han83,Man94], ized and although the various phases have been improved over the Bézier clipping [NSK90, EHS05], and numerical techniques [Tot85, years, the result is not perfect yet. JB86, AGM06]. The most visible artifacts resulting from the fabrication process In optics, much work addressed intersections with spherical sur- are due to a misalignment between grinding and optical axis, grind- faces but few approaches covered aspheric lenses; typically, tessel- ing asymmetries resulting from the swinging of the tools, and limited lation [MNN∗10] and Delaunay triangles [OSRM09]) or Newton- grinding precision. A few nanometers of deviation can lead to a visi- Raphson methods [AS52,For66] were applied. Tessellation is costly ble impact on bokeh. We model the resulting imperfections indirectly for sufficient precision, while the Newton-Raphson method is fast by simulating the grinding process itself. Following the fabrication

© 2016 The Author(s) Computer Graphics Forum © 2016 The Eurographics Association and John Wiley & Sons Ltd. H. Joo & S. Kwon & S. Lee & E. Eisemann & S. Lee / Efﬁcient Ray Tracing Through Aspheric Lenses and Imperfect Bokeh Synthesis

(a) grinding (b) polishing Offline preprocessing Online rendering

load an optical system for each RGB channel swinging swinging for each ray for each surface element s if s is an aperture apply aperture stop at intersection else if s is an aspheric lens find analytical intersection with proxies for each aspheric lens test numerical intersection - find proxy surfaces refract ray (with jittered normal) else if s is a spherical lens Figure 3: Simple mechanics of lens fabrication via a grinder. test analytical intersection

virtual grinder refract ray - generate bump maps triangulate and rasterize rays - generate normal maps apply FrFT

accumulate the result

+ + polish Grind Grind Figure 5: Overview of the rendering pipeline of our system.

White noise

5.1. Ray-tracing an Aspheric Lens Bump map Normal map To test the intersection of a ray and the aspheric surface, we use a Figure 4: Example of a grinding pattern generation; parameters bracketing-based root-finding method (either the bisection or the are 5000× exaggerated for illustration purposes. false position method). The explicit lens definition (Eq. (1)) is incon- venient for this process, and we instead reformulate it to an implicit representation. method, we create grinding patterns by superposing circles with a Eq. (1) defines the height-field lens surface given a radial co- smooth boundary (reflecting the grinding head) and decreasing radii ordinate r, where r is shared by several points located around the (or a spiral, depending on the simulated tool). We apply a perturba- optical axis (z-axis). For an arbitrary 3D point v (with its Carte- tion via a wave pattern, whose strength is user controlled. Similarly, sian coordinates vx, vy, and vz) this radial coordinate is expressed 2 2 1/2 following real-world examples, the result involves between 50 to as r := (vx + vy) . When v lies on the lens surface, vz = z(r) is 120 rings. Each ring is also drawn with a low-frequency jittering to satisfied. This leads to the following implicit formulation: simulate the swinging. The rings are accumulated to create a height 2 2 1/2 f (v) := z((v + v ) ) − vz, (2) profile and combined with a white noise texture to simulate general x y imperfections. The final result is a height field from which we derive where the sign of f (v) implies if a point lies above or below the a normal map. The latter is used during rendering to influence the aspheric-lens surface and the aspheric-lens surface is the set {v ∈ 3 refractive directions. The process is illustrated in Figure4. R | f (v) = 0}. One more artifact, which can be independent of the production When inserting a ray equation into the implicit formulation, process, are tiny dirt marks. These can become visible with large the problem becomes a root-finding process. Derivative-based out-of-focus bokeh. Their appearance can be arbitrary, and we model methods exist, which utilize the base curvature as an initial them as noisy granular details using a noise generator and augment guess [AS52, For66, Smi00]. However, the use of gradients can the previously-derived texture. Further, we store this pattern also as be difficult and unreliable, as many GPUs only have limited or no an additional color channel to reduce the intensity of intersecting support for double precision. Our solution instead relies on a bracket- rays. ing method, which does not require to evaluate gradients. Avoiding double-precision computations on GPUs leads to a great performance gain; such preference for bracketing can be found in various 5. Rendering GPU approaches [BES12] and is a robust and efficient choice. Given the lens model, we will, similarly to previous work [KMH95, The bisection method is the simpler method of the two and initial- SDHL11, HESL11], apply a deterministic forward ray tracing to ized with a tight interval. It is iteratively subdivided at its midpoint, trace the light propagation in an optical system. An overview of the yielding two subintervals. The process always continues with the various steps of our approach is given in Figure5. The incoming subinterval whose extremities lie on opposite sides of the surface. rays are generated at the entrance pupil. When the rays encounter The false position method [BF01] is similar but assumes in each a lens element, we test for intersection. If the element is missed or iteration that the function is linear between the two interval extremi- a total reflection occurs, we stop the ray traversal. At the aperture, ties. The root of this linear function defines the position to produce we use a polygonal texture to define the iris and to test if rays are subintervals. blocked [HESL11]. Once a ray reaches the sensor, we accumulate its irradiance via additive alpha blending. Proxy Geometry The efficiency of our root-finding process de- To account for imperfections, we apply the normal maps, which pends on the initial interval, which we initialize by determining the are stored as 2D textures for each aspheric surface. Based on the ray intersection with a tight surface-englobing proxy geometry. In texture lookup, the intersecting ray is jittered. As the normal vectors principle, one could define two planes that englobe the lens surface, are based on our virtual grinding patterns, the result will exhibit the but seen that many aspheric lenses are based on a spherical surface, typical bokeh with ring patterns, often called “onion-ring” bokeh. two spherical interfaces usually deliver a tighter fit.

z z F0[f] F0.2[f] F0.4[f] F0.6[f] F0.8[f] F1.0[f]

(a) r (b) r sa sa Figure 6: Example deﬁnition of pairs of proxy surfaces: (a) spheres and (b) planes. Figure 7: Evolution of the fractional Fourier transform along its order α ∈ [0,1] for a heptagonal aperture.

The proxy geometry is determined in an offline preprocess. First, evaluation when the aperture is traversed. As the visual differences we construct a spherical interface determined by the surface point are typically low, we propose to choose α empirically. on the optical axis and the intersection between the tube and the lens (in many optical data sheets, the corresponding radius is given 5.3. Anti-Aliased and Spectral Rendering as sa). Next, we find the two tangent positions to the aspheric lens surface along the z-axis. These two configurations define our proxy Our solution supports similar extensions as one by Hullin et geometry (Figure6). The tangent position could be determined via a al. [HESL11]. To avoid noise, we trace a beam of three rays and root finding (inserting their expression in Eq. (2)) but would need to rasterize the resulting triangle with interpolation on the sensor. Addi- be coupled to an additional test of the lens boundary. In consequence, tionally, we employ multisample antialiasing (MSAA). For spectral for simplicity and to support even more general lens surfaces, we rendering, we compute different refractive indices depending on the use a sample-based procedure. It proved sufficient in practice and wavelength [HESL11,SDHL11] and trace the result of each spectral tests different configurations by evaluating a dense set of surface sample separately. points for tangency (all samples positive/negative and one sample close/equal to zero). 6. Results and Discussions We implemented our system using the OpenGL API on an Intel Core 5.2. Fractional Diffraction Synthesis i7 machine with NVIDIA GTX 980 Ti. Four patented lenses were Light patterns (or its lens-flare ghosts) passing through an aper- chosen for our experiments (see Figure8 for illustrations, patent ture usually exhibit ringing at edges, which is crucial for a re- numbers, and the corresponding output generated by our approach). alistic appearance of bokeh patterns (Figure7). This ringing is The red-colored interfaces indicate aspheric elements. Normal maps known to be caused by near-field Fresnel diffraction [Kin75]. This are applied to the aspheric surfaces only because the manufacturing Fresnel diffraction can be formulated to a fractional Fourier Trans- imperfections are uncommon for spherical surfaces. In combination form (FrFT) [PF94], which uses a fractional order α to control the with the FrFT, the bokeh renderings appear similar to photographs. amount of transition to the full Fourier Transform (FT); see Fig- Our algorithm performs the bokeh synthesis in a two-step process; ure7. The FrFT Fα(u) of a signal g(x) is formulated as: Fα(u) := an offline stage (the virtual grinding to produce the normal map, R ∞ −∞ Kα(u,x)g(x)dx, where the transform kernel Kα(u,x) is: and the proxy-surface fitting on the CPU) and an online stage on the GPU (ray tracing through the lens system, rasterization into p −2π j(cscα ux−cotα (u2+x2)/2) Kα(u,x)= 1− j cotα e . (3) the sensor texture, and the FrFT for starburst streaks). Note that the offline preprocessing is light-weight requiring only 60 ms on We implement a discrete FrFT on the GPU, instead of the CPU average over all shown camera models. Table1 summarizes the (as done by the previous studies [OZK01, HESL11]); see the ac- rendering performance during the online stage using 5122 and 10242 companying supplement for details. Since the phase components resolutions for the entering rays on the entrance pupil and the sensor of FrFT (see Eq. (3)) shift much faster than FT, it requires a denser plane. We measured the result for the synthesis of an achromatic sampling to avoid aliasing. The typical implementation relying on (single wavelength) bokeh resulting from a directional light source. fast FT (FFT) is slow and needs considerable zero padding, which may not fit into GPU memory. So, we take a direct discrete sam- Overall, our system is capable of real-time performance, even pling approach. Despite its several limitations (e.g., non-unitary when applying FrFT. Ray tracing and rasterization scale linearly and irreversible [PD00]), it is a good GPU fit. In particular, the with the complexity of the lens systems and the number of rays, denser sampling can make use of the built-in bilinear interpolation while the FrFT scales with the resolution. FrFT is the most significant bottleneck, but still up to 20× faster than existing Matlab without explicit upscaling, which greatly improves performance and 2 addresses the major bottleneck of our pipeline. implementations (e.g., 2.8 s at 1024 ). For dispersion, we need to repeat the achromatic processing for each wavelength, which lin- We choose α empirically (typically, α ∈ [0.1,0.2]), which gives early scales with the number of spectral samples; e.g., computing visually-plausible results. The concrete value is chosen to best match with seven wavelengths takes around 1.5 s at a 10242 resolution. the appearance of a real photograph. The reason being that, albeit conceptually well defined [PF94], an efficient and accurate compu- 6.1. Defocus Blur and Lens Flares tation of α is a significant challenge. We would need to not only evaluate per-object αs, but also composite FrFTs of different αs; Similarly to previous work, we can use our model to compute de- for multi-lens systems, each bounce would need an additional FrFT focus blur and lens flares, but even with our efficient ray-tracing

System 1 (USP3992085) α = 0.1 α = 0.1

System 2 (USP4514052) α = 0.1 α = 0.1

System 3 (USP5793533) α = 0.01 α = 0.01

System 4 (USP5805359) α = 0.12 α = 0.12

Figure 8: Examples of lens systems (yellow, red, and green elements for the entrance pupil, aspheric elements, and sensor planes, respectively), normal maps (exaggerated for legibility) and their corresponding bokeh textures.

Table 1: Performance of bokeh rendering measured in frame time Figure9 illustrates an example of thin-lens depth-of-field blur (ms) for the four lens systems with respect to the number of rays (ray using accumulation buffering [HA90] in combination with a pre- tracing and raster) and resolution (FrFT). computed bokeh texture. Each rendered view is modulated by the intensity stored in the bokeh texture. We used a 2562 resolution in System ID Ray tracing Raster FrFT the accompanying video and 5122 in the figure. In practice, 2562 5122 10242 5122 10242 5122 10242 usually suffices. The accumulation buffering remains a costly process (e.g., 100 s for 2562 samples, even though a single sample 1 0.77 2.43 0.26 0.59 26.93 199.14 only takes 1.5 ms) but it would be possible to rely on faster meth- 2 1.14 3.97 0.43 0.79 28.04 202.30 ods [LES09, LES10]. 3 0.87 2.76 0.24 0.59 27.18 202.11 4 0.65 2.10 0.29 0.62 27.17 201.18 Similarly, we can approximate lens flare. Real-time lens-flare rendering typically uses the same type of pre-defined texture, which illustrates the light rays traversing the aperture. We used a recent sprite-based lens-flare rendering [LE13] for demonstration purposes solution, it would be very costly to apply our approach for full (Figure 10). It is based on the paraxial linear approximation to locate depth-of-field or lens-flare rendering on entire scenes. It would re- the flares in real time. quire a large number (e.g., thousands) of lens samples, similarly to glossy highlights in Monte-Carlo rendering and there are no good solutions yet for this particular case, such as adaptive sampling or 7. Conclusion and Discussions interpolation methods. Consequently, the ray tracing scheme and the fractional Fourier transforms can become bottlenecks. One pos- We presented an efficient ray-tracing technique for parametric as- sibility for higher efficiency is to generate a bokeh texture with our pheric lenses. Our solution includes the simulation of mechanical system, which is then used in a simplified simulation. imperfections, which builds upon the lens-fabrication process and

(a) (b) (c)

Figure 9: Examples of defocus blur [HA90] using bokeh textures generated with our solution.

(a) (b) (c)

Figure 10: Examples of a texture sprite-based lens-ﬂare rendering [LE13] to which our bokeh textures applied.

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